MTH 201 Final, Spring 2009

Calc III SampekFinal Exam Spring 09 . Evaluate the following limits: (Show your work) a\? (a) lim ( + 2) 20+ x (b) lim yn n—00 . 3\* © jim (1+) . Evaluate SH wnere y = (23 + 5a? — 3x)5(/e+2)8 . Evaluate the following integrals: (a) / ze~*dx 1 1 (b) lamer” . Determine whether the following improper integrals converge or diverge. Justify your answer! (a) [setae (b) f aS ae 3 de () \ome . Which of the following sequences {a,,}converge or diverge. In case of convergence, find the limit: 2 (a) On = ee Ont — 2n+Inn—-3 sinn (b) aq = (-1). 2" . Determine whether the following series converge or diverge. In case of an alternating series, determine if the series converges absolutely, conditionally or diverges: 0 7. For what values of x does the series So 2r(@ — 1)” converge: n=1 1 8. (a) Find the Maclaurin series for f(x) = To’ where |x] <1 (b) Deduce the series representation for In|1 — z| (c) Deduce an approximation for In(3/4). (d) Deduce from (a) the derivative f{)(0) of order12 of the function above at 0. 9. Determine if the following improper integrals converge or diverge: (Explain why they are improper) oe 1 (2) I, eri” 2 Bo b (b) oz? —1 oO 4 () f fe nde dx 10. Determine whether the following sequences{a,,} converge or diverge: (@) a9 = Oo (b) an = (1 = 2" 11. If the sequences {a,} and {bn} converge to A and B respectively, what can you conclude about the convergence or divergence of the following sequences? (Justify your answer if true or give a counterexample if false) (a) {a2 - bn} 0) + nm 12. Determine whether the following series converge or diverge: @) SOs ore) =. vn? +1 (c) > (3 - = i} 0° oo 13, If Soon converges to A and Von converges to B, what can you say about the convergence or divergence n=l n=1 of the series: (Justify your answer if true or give a counterexample if false) (0) 37 ay — 3>n) (b) $0 (@n +1) wo nm 14. For what values of x does the series converge » Ge *) n=1 converge? 15. Find the McLaurin series for cos(x) 16. Find a series representation for / e-* da. . sing/x~1 17. Evaluate, using power series the limit below: lim sna/eat 2 x (a) Find the domain and range of f. (b) Describe the level curves for z = 0, 1,1/2 (c) Determine if lim __ f(x,y) exists. Justify your answer 18. Given the function z = f(x,y) = (2,y)> (0,0) _ , Ow, 2 . 19. Find Or ifw = (z+y+z)*, where c =r —s, y = cos(r +8), and z= sin(r + s). . 1 20. Consider the surface (S) : z = f(z,y) = Pay (a) Find the rate of change of z at the point (1,1,1/2),in the direction of the vector Aza=T+ 7 (b) Write the equation of the plane tangent to (5) at (1, 1,1/2) (c) In which direction does z change most rapidly at (1, 1,1/2)? (d) Write the equation of the line (Z) normal to (S) at (1,1, 1/2). 21. Find the volume of the solid bounded above by the surface z = x? — y® and below by the rectangle 2<a<5,3<y<7. 4 2 on? 22. Evaluate the integral: | / © dedy.(Hint: reverse the order of integration.) ot wavy